let s be natural Number ; :: thesis: s ! > 0
A0: s is Nat by TARSKI:1;
defpred S₁[ Nat] means $1 ! > 0 ;

A1: now :: thesis: for s being Nat st S₁[s] holds
S₁[s + 1]

let s be Nat; :: thesis: ( S₁[s] implies S₁[s + 1] )
assume S₁[s] ; :: thesis: S₁[s + 1]
then (s + 1) * (s !) > (s + 1) * 0 ;
hence S₁[s + 1] by Th15; :: thesis: verum

end;

A2: S₁[ 0 ] by RVSUM_1:94;
for s being Nat holds S₁[s] from NAT_1:sch 2(A2, A1);
hence s ! > 0 by A0; :: thesis: verum